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Quaternion.h
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1 // @(#)root/mathcore:$Id$
2 // Authors: W. Brown, M. Fischler, L. Moneta 2005
3 
4  /**********************************************************************
5  * *
6  * Copyright (c) 2005 , LCG ROOT FNAL MathLib Team *
7  * *
8  * *
9  **********************************************************************/
10 
11 // Header file for rotation in 3 dimensions, represented by a quaternion
12 // Created by: Mark Fischler Thurs June 9 2005
13 //
14 // Last update: $Id$
15 //
16 #ifndef ROOT_Math_GenVector_Quaternion
17 #define ROOT_Math_GenVector_Quaternion 1
18 
19 
26 
27 #include <algorithm>
28 #include <cassert>
29 
30 
31 namespace ROOT {
32 namespace Math {
33 
34 
35 //__________________________________________________________________________________________
36  /**
37  Rotation class with the (3D) rotation represented by
38  a unit quaternion (u, i, j, k).
39  This is the optimal representation for multiplication of multiple
40  rotations, and for computation of group-manifold-invariant distance
41  between two rotations.
42  See also ROOT::Math::AxisAngle, ROOT::Math::EulerAngles, and ROOT::Math::Rotation3D.
43 
44  @ingroup GenVector
45  */
46 
47 class Quaternion {
48 
49 public:
50 
51  typedef double Scalar;
52 
53  // ========== Constructors and Assignment =====================
54 
55  /**
56  Default constructor (identity rotation)
57  */
59  : fU(1.0)
60  , fI(0.0)
61  , fJ(0.0)
62  , fK(0.0)
63  { }
64 
65  /**
66  Construct given a pair of pointers or iterators defining the
67  beginning and end of an array of four Scalars
68  */
69  template<class IT>
70  Quaternion(IT begin, IT end) { SetComponents(begin,end); }
71 
72  // ======== Construction From other Rotation Forms ==================
73 
74  /**
75  Construct from another supported rotation type (see gv_detail::convert )
76  */
77  template <class OtherRotation>
78  explicit Quaternion(const OtherRotation & r) {gv_detail::convert(r,*this);}
79 
80 
81  /**
82  Construct from four Scalars representing the coefficients of u, i, j, k
83  */
85  fU(u), fI(i), fJ(j), fK(k) { }
86 
87  // The compiler-generated copy ctor, copy assignment, and dtor are OK.
88 
89  /**
90  Re-adjust components to eliminate small deviations from |Q| = 1
91  orthonormality.
92  */
93  void Rectify();
94 
95  /**
96  Assign from another supported rotation type (see gv_detail::convert )
97  */
98  template <class OtherRotation>
99  Quaternion & operator=( OtherRotation const & r ) {
100  gv_detail::convert(r,*this);
101  return *this;
102  }
103 
104  // ======== Components ==============
105 
106  /**
107  Set the four components given an iterator to the start of
108  the desired data, and another to the end (4 past start).
109  */
110  template<class IT>
111 #ifndef NDEBUG
112  void SetComponents(IT begin, IT end) {
113 #else
114  void SetComponents(IT begin, IT ) {
115 #endif
116  fU = *begin++;
117  fI = *begin++;
118  fJ = *begin++;
119  fK = *begin++;
120  assert (end==begin);
121  }
122 
123  /**
124  Get the components into data specified by an iterator begin
125  and another to the end of the desired data (4 past start).
126  */
127  template<class IT>
128 #ifndef NDEBUG
129  void GetComponents(IT begin, IT end) const {
130 #else
131  void GetComponents(IT begin, IT ) const {
132 #endif
133  *begin++ = fU;
134  *begin++ = fI;
135  *begin++ = fJ;
136  *begin++ = fK;
137  assert (end==begin);
138  }
139 
140  /**
141  Get the components into data specified by an iterator begin
142  */
143  template<class IT>
144  void GetComponents(IT begin ) const {
145  *begin++ = fU;
146  *begin++ = fI;
147  *begin++ = fJ;
148  *begin = fK;
149  }
150 
151  /**
152  Set the components based on four Scalars. The sum of the squares of
153  these Scalars should be 1; no checking is done.
154  */
156  fU=u; fI=i; fJ=j; fK=k;
157  }
158 
159  /**
160  Get the components into four Scalars.
161  */
162  void GetComponents(Scalar & u, Scalar & i, Scalar & j, Scalar & k) const {
163  u=fU; i=fI; j=fJ; k=fK;
164  }
165 
166  /**
167  Access to the four quaternion components:
168  U() is the coefficient of the identity Pauli matrix,
169  I(), J() and K() are the coefficients of sigma_x, sigma_y, sigma_z
170  */
171  Scalar U() const { return fU; }
172  Scalar I() const { return fI; }
173  Scalar J() const { return fJ; }
174  Scalar K() const { return fK; }
175 
176  // =========== operations ==============
177 
178  /**
179  Rotation operation on a cartesian vector
180  */
182  XYZVector operator() (const XYZVector & v) const {
183 
184  const Scalar alpha = fU*fU - fI*fI - fJ*fJ - fK*fK;
185  const Scalar twoQv = 2*(fI*v.X() + fJ*v.Y() + fK*v.Z());
186  const Scalar twoU = 2 * fU;
187  return XYZVector ( alpha * v.X() + twoU * (fJ*v.Z() - fK*v.Y()) + twoQv * fI ,
188  alpha * v.Y() + twoU * (fK*v.X() - fI*v.Z()) + twoQv * fJ ,
189  alpha * v.Z() + twoU * (fI*v.Y() - fJ*v.X()) + twoQv * fK );
190  }
191 
192  /**
193  Rotation operation on a displacement vector in any coordinate system
194  */
195  template <class CoordSystem,class Tag>
198  DisplacementVector3D< Cartesian3D<double> > xyz(v.X(), v.Y(), v.Z());
201  vNew.SetXYZ( rxyz.X(), rxyz.Y(), rxyz.Z() );
202  return vNew;
203  }
204 
205  /**
206  Rotation operation on a position vector in any coordinate system
207  */
208  template <class CoordSystem, class Tag>
213  return PositionVector3D<CoordSystem,Tag> ( rxyz );
214  }
215 
216  /**
217  Rotation operation on a Lorentz vector in any 4D coordinate system
218  */
219  template <class CoordSystem>
223  xyz = operator()(xyz);
224  LorentzVector< PxPyPzE4D<double> > xyzt (xyz.X(), xyz.Y(), xyz.Z(), v.E());
225  return LorentzVector<CoordSystem> ( xyzt );
226  }
227 
228  /**
229  Rotation operation on an arbitrary vector v.
230  Preconditions: v must implement methods x(), y(), and z()
231  and the arbitrary vector type must have a constructor taking (x,y,z)
232  */
233  template <class ForeignVector>
234  ForeignVector
235  operator() (const ForeignVector & v) const {
238  return ForeignVector ( rxyz.X(), rxyz.Y(), rxyz.Z() );
239  }
240 
241  /**
242  Overload operator * for rotation on a vector
243  */
244  template <class AVector>
245  inline
246  AVector operator* (const AVector & v) const
247  {
248  return operator()(v);
249  }
250 
251  /**
252  Invert a rotation in place
253  */
254  void Invert() { fI = -fI; fJ = -fJ; fK = -fK; }
255 
256  /**
257  Return inverse of a rotation
258  */
259  Quaternion Inverse() const { return Quaternion(fU, -fI, -fJ, -fK); }
260 
261  // ========= Multi-Rotation Operations ===============
262 
263  /**
264  Multiply (combine) two rotations
265  */
266  /**
267  Multiply (combine) two rotations
268  */
270  return Quaternion ( fU*q.fU - fI*q.fI - fJ*q.fJ - fK*q.fK ,
271  fU*q.fI + fI*q.fU + fJ*q.fK - fK*q.fJ ,
272  fU*q.fJ - fI*q.fK + fJ*q.fU + fK*q.fI ,
273  fU*q.fK + fI*q.fJ - fJ*q.fI + fK*q.fU );
274  }
275 
276  Quaternion operator * (const Rotation3D & r) const;
277  Quaternion operator * (const AxisAngle & a) const;
278  Quaternion operator * (const EulerAngles & e) const;
279  Quaternion operator * (const RotationZYX & r) const;
280  Quaternion operator * (const RotationX & rx) const;
281  Quaternion operator * (const RotationY & ry) const;
282  Quaternion operator * (const RotationZ & rz) const;
283 
284  /**
285  Post-Multiply (on right) by another rotation : T = T*R
286  */
287  template <class R>
288  Quaternion & operator *= (const R & r) { return *this = (*this)*r; }
289 
290 
291  /**
292  Distance between two rotations in Quaternion form
293  Note: The rotation group is isomorphic to a 3-sphere
294  with diametrically opposite points identified.
295  The (rotation group-invariant) is the smaller
296  of the two possible angles between the images of
297  the two totations on that sphere. Thus the distance
298  is never greater than pi/2.
299  */
300 
301  Scalar Distance(const Quaternion & q) const ;
302 
303  /**
304  Equality/inequality operators
305  */
306  bool operator == (const Quaternion & rhs) const {
307  if( fU != rhs.fU ) return false;
308  if( fI != rhs.fI ) return false;
309  if( fJ != rhs.fJ ) return false;
310  if( fK != rhs.fK ) return false;
311  return true;
312  }
313  bool operator != (const Quaternion & rhs) const {
314  return ! operator==(rhs);
315  }
316 
317 private:
318 
323 
324 }; // Quaternion
325 
326 // ============ Class Quaternion ends here ============
327 
328 /**
329  Distance between two rotations
330  */
331 template <class R>
332 inline
333 typename Quaternion::Scalar
334 Distance ( const Quaternion& r1, const R & r2) {return gv_detail::dist(r1,r2);}
335 
336 /**
337  Multiplication of an axial rotation by an AxisAngle
338  */
339 Quaternion operator* (RotationX const & r1, Quaternion const & r2);
340 Quaternion operator* (RotationY const & r1, Quaternion const & r2);
341 Quaternion operator* (RotationZ const & r1, Quaternion const & r2);
342 
343 /**
344  Stream Output and Input
345  */
346  // TODO - I/O should be put in the manipulator form
347 
348 std::ostream & operator<< (std::ostream & os, const Quaternion & q);
349 
350 
351 } // namespace Math
352 } // namespace ROOT
353 
354 #endif // ROOT_Math_GenVector_Quaternion
Class describing a generic LorentzVector in the 4D space-time, using the specified coordinate syste...
Definition: LorentzVector.h:59
Scalar I() const
Definition: Quaternion.h:172
double dist(Rotation3D const &r1, Rotation3D const &r2)
Definition: 3DDistances.cxx:48
DisplacementVector3D< CoordSystem, Tag > & SetXYZ(Scalar a, Scalar b, Scalar c)
set the values of the vector from the cartesian components (x,y,z) (if the vector is held in polar or...
Namespace for new ROOT classes and functions.
Definition: StringConv.hxx:21
Rotation class representing a 3D rotation about the Z axis by the angle of rotation.
Definition: RotationZ.h:43
Scalar J() const
Definition: Quaternion.h:173
Class describing a generic position vector (point) in 3 dimensions.
Rotation class with the (3D) rotation represented by angles describing first a rotation of an angle p...
Definition: RotationZYX.h:61
Quaternion & operator=(OtherRotation const &r)
Assign from another supported rotation type (see gv_detail::convert )
Definition: Quaternion.h:99
Rotation class with the (3D) rotation represented by a unit quaternion (u, i, j, k).
Definition: Quaternion.h:47
#define R(a, b, c, d, e, f, g, h, i)
Definition: RSha256.hxx:110
std::ostream & operator<<(std::ostream &os, const AxisAngle &a)
Stream Output and Input.
Definition: AxisAngle.cxx:91
AxisAngle class describing rotation represented with direction axis (3D Vector) and an angle of rotat...
Definition: AxisAngle.h:41
Scalar X() const
Cartesian X, converting if necessary from internal coordinate system.
Quaternion Inverse() const
Return inverse of a rotation.
Definition: Quaternion.h:259
void GetComponents(IT begin, IT end) const
Get the components into data specified by an iterator begin and another to the end of the desired dat...
Definition: Quaternion.h:129
Rotation class representing a 3D rotation about the Y axis by the angle of rotation.
Definition: RotationY.h:43
Scalar U() const
Access to the four quaternion components: U() is the coefficient of the identity Pauli matrix...
Definition: Quaternion.h:171
Scalar Z() const
Cartesian Z, converting if necessary from internal coordinate system.
Scalar Y() const
Cartesian Y, converting if necessary from internal coordinate system.
Quaternion(const OtherRotation &r)
Construct from another supported rotation type (see gv_detail::convert )
Definition: Quaternion.h:78
void SetComponents(Scalar u, Scalar i, Scalar j, Scalar k)
Set the components based on four Scalars.
Definition: Quaternion.h:155
Class describing a generic displacement vector in 3 dimensions.
ROOT::R::TRInterface & r
Definition: Object.C:4
Quaternion(Scalar u, Scalar i, Scalar j, Scalar k)
Construct from four Scalars representing the coefficients of u, i, j, k.
Definition: Quaternion.h:84
auto * a
Definition: textangle.C:12
void GetComponents(Scalar &u, Scalar &i, Scalar &j, Scalar &k) const
Get the components into four Scalars.
Definition: Quaternion.h:162
Rotation class representing a 3D rotation about the X axis by the angle of rotation.
Definition: RotationX.h:43
Rotation class with the (3D) rotation represented by a 3x3 orthogonal matrix.
Definition: Rotation3D.h:65
void GetComponents(IT begin) const
Get the components into data specified by an iterator begin.
Definition: Quaternion.h:144
DisplacementVector3D< Cartesian3D< double >, DefaultCoordinateSystemTag > XYZVector
Rotation operation on a cartesian vector.
Definition: Quaternion.h:181
void convert(R1 const &, R2 const)
Definition: 3DConversions.h:41
EulerAngles class describing rotation as three angles (Euler Angles).
Definition: EulerAngles.h:43
you should not use this method at all Int_t Int_t Double_t Double_t Double_t e
Definition: TRolke.cxx:630
Namespace for new Math classes and functions.
bool operator!=(const Quaternion &rhs) const
Definition: Quaternion.h:313
void Rectify()
Re-adjust components to eliminate small deviations from |Q| = 1 orthonormality.
Definition: Quaternion.cxx:35
Quaternion & operator*=(const R &r)
Post-Multiply (on right) by another rotation : T = T*R.
Definition: Quaternion.h:288
void Invert()
Invert a rotation in place.
Definition: Quaternion.h:254
Scalar Distance(const Quaternion &q) const
Distance between two rotations in Quaternion form Note: The rotation group is isomorphic to a 3-spher...
Definition: Quaternion.cxx:92
bool operator==(const Quaternion &rhs) const
Equality/inequality operators.
Definition: Quaternion.h:306
Scalar K() const
Definition: Quaternion.h:174
DefaultCoordinateSystemTag Default tag for identifying any coordinate system.
float * q
Definition: THbookFile.cxx:87
Rotation3D::Scalar Scalar
void SetComponents(IT begin, IT end)
Set the four components given an iterator to the start of the desired data, and another to the end (4...
Definition: Quaternion.h:112
Quaternion()
Default constructor (identity rotation)
Definition: Quaternion.h:58
AVector operator*(const AVector &v) const
Overload operator * for rotation on a vector.
Definition: Quaternion.h:246
XYZVector operator()(const XYZVector &v) const
Definition: Quaternion.h:182
Quaternion(IT begin, IT end)
Construct given a pair of pointers or iterators defining the beginning and end of an array of four Sc...
Definition: Quaternion.h:70